Consider the following null and alternative hypotheses: Ho: p= 0.70 H1: p? 0.70 A sample of size n = 100 is to be taken and the sample proportion p calculated from the sample observations. The decision rule for this test has been established as: Reject Ho if p < 0.60 or if p > 0.80 Do not reject Ho if 0.60 ? p ? 0.80. a. What is the significance level ? for this test? b. Suppose that p is really 0.85. What is the probability of committing a type II error, not rejecting a false null hypothesis for this decision rule?
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Since the sample size is 100, we can use the normal distribution to approximate the sampling distribution of the sample proportion. The standard deviation of the sampling distribution is given by: $$ \sigma_p = \sqrt{\frac{p(1-p)}{n}} = Show more…
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Consider Example $9.6$ with the null hypothesis $$ H_{0}: P=P_{0}=0.50 $$ and the alternative hypothesis $$ H_{0}: P \neq 0.50 $$ The decision rule is $$ \begin{gathered} \frac{\hat{p}_{x}-0.50}{\sqrt{0.50(1-0.50) / 600}}<-1.96 \text { or } \\ \frac{\hat{p}_{x}-0.50}{\sqrt{0.50(1-0.50) / 600}}>1.96 \end{gathered} $$ with a sample size of $n=600$. What is the probability of Type II error if the actual population proportion is each of the following? a. $P=0.52$ b. $P=0.58$ c. $P=0.53$ d. $P=0.48$. e. $P=0.43$
For the hypotheses below, test = 0.025 with n = 100 and p = 0.69. State: a. the decision rule in terms of the critical value of the test statistic. b. the calculated value of the test statistic. c. the conclusion. Ho: p = 0.74 HA: p < 0.74 a. This is a test of the population proportion. The decision rule is to reject the null hypothesis if the calculated value of the test statistic, z, is less than the critical value. Otherwise, do not reject the null hypothesis. (Round to two decimal places as needed.) The critical value, z =
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Consider the following hypothesis test. Ho: rho = 0.5 Ha: rho not equals 0.5 A sample of 800 is provided and produces a sample proportion of 0.58. Using alpha = 0.05, what is the rejection rule? A. Reject Ho if z < 1.645. B. Reject Ho if z < -1.96 or if z > 1.96. C. Reject Ho if z > 1.96. D. Accept Ha if z < -1.645 What is the standard error of the proportion? A. 0.01549 B. 0.01767 C. 0.01127 D. 0.02283 What is the value of the test statistic? What should you conclude? A. -4.5; reject Ho. B. 2.33; reject Ho. C. 1.58; do not reject Ho. D. 4.52; reject Ho.
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